A framework for addressing a potential instability problem in adaptive control and iterative identification and controller design algorithms is proposed. Suppose an unknown plant is stabilized by a known controller, some knowledge of this stable closed-loop system is available, and the use of a new controller to replace the current stabilizing controller becomes imminent. Our analysis results assume that the `unknown' plant and the controllers are all nonlinear. We further develop a data-based test which utilizes a limited amount of experimental data from an existing stable closed-loop (a plant in connection with a linear stabilizing controller) for verifying that the introduction of a new nonlinear controller will stabilize the unknown plant.